The superficial similarity between rumours and epidemics breaks down on closer scrutiny; a feature peculiar to the rumour-spreading situation leads to striking qualitative differences in the behaviour of the two phenomena whether one uses a stochastic model or the associated deterministic model. A preliminary account is given here of a new procedure, “the principle of the diffusion of arbitrary constants”, which can be used to study the variance of the fluctuations of the sample trajectory in the stochastic model about the unique trajectory in the associated deterministic approximation. Numerical evidence (based on Monte Carlo and other calculations) is given to illustrate the effectiveness of the “principle” in the present application.